Mathematics

Problem

Define

H(x)=\int_{0}^{x^3} \sin(t^2)\,dt-\int_{1}^{x} \sin(t^2)\,dt,

for all real numbers

x

for which the integrals are defined. (a) Find

H'(x)

. Clearly indicate how each term in your answer arises. (b) Compute

H'(1)

. Is

H

increasing or decreasing at

x=1

? Justify your conclusion.